Projection Algorithms for Variational Inclusions

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality

Quasi Variational Inclusions Involving Three Operators

In this paper, we consider some new classes of the quasi-variational inclusions involving three monotone operators. Some interesting problems such as variational inclusions involving sum of two monotone operators, difference of two monotone operators, system of absolute value equations, hemivariational inequalities and variational inequalities are the special cases of quasi variational inequalities. It is shown that quasi-variational inclusions are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inclusions and related optimization problems are suggested by using resolvent methods, resolvent equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results

Quasi Variational Inclusions Involving Three Operators

In this paper, we consider some new classes of the quasi-variational inclusions involving three monotone operators. Some interesting problems such as variational inclusions involving sum of two monotone operators, difference of two monotone operators, system of absolute value equations, hemivariational inequalities and variational inequalities are the special cases of quasi variational inequalities. It is shown that quasi-variational inclusions are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inclusions and related optimization problems are suggested by using resolvent methods, resolvent equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results

A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given

Inertial Douglas-Rachford splitting for monotone inclusion problems

We propose an inertial Douglas-Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigate its convergence properties. To this end we formulate first the inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating the set of fixed points of a nonexpansive operator, for which we also provide an exhaustive convergence analysis. By using a product space approach we employ these results to the solving of monotone inclusion problems involving linearly composed and parallel-sum type operators and provide in this way iterative schemes where each of the maximally monotone mappings is accessed separately via its resolvent. We consider also the special instance of solving a primal-dual pair of nonsmooth convex optimization problems and illustrate the theoretical results via some numerical experiments in clustering and location theory.Comment: arXiv admin note: text overlap with arXiv:1402.529